The geometry Anosov subgroups in Lie groups

李群中的几何阿诺索夫子群

• 批准号: RGPIN-2020-05557
• 负责人: Burelle, JeanPhilippe
• 金额: $ 1.31万
• 依托单位: Université de Sherbrooke
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740653
• 关键词: geometry; Anosov; subgroups; Lie; groups

The field of locally homogeneous geometric structures on manifolds generalizes the field of cartography, in the following sense : drawing an atlas of the world is like trying to describe the shape of the earth using flat geometry, and a locally homogeneous geometric structure is like trying to describe the shape of a more general world (or manifold) using some homogeneous geometry. One of the first theorems in the subject states that it is in fact impossible to construct a complete atlas of the world without distortion, that is, that there exists no Euclidean (flat) structure on the two-dimensional sphere. Already in the study of flat structures, the Bieberbach theorems exemplify the tight relationship between geometric structures and groups of symmetries. These theorems imply that, up to finite covers, the only finite-area flat structures in 2 dimensions are similar to the world of Pac-man, that is, they are obtained by identifying parallel edges of a parallelogram. It is no coincidence that the parallelogram can tile the plane by translating multiple copies of itself, creating a tiling with translational symmetry. The study of symmetries between objects or concepts is foundational to a range of topics in pure and applied mathematics and in other sciences. We say that an object has continuous symmetry if its symmetries can be very small motions, for instance a very small rotation of a sphere is a symmetry, so it has continuous symmetry. We say that it has discrete symmetry if this is not the case, for instance a cube has discrete symmetry because it only has 90 degree rotational symmetry. The symmetries of the cube are a subset of the symmetries of the sphere. Similarly, the plane has continuous symmetry but a parallelogram tiling has discrete symmetry within the symmetries of the plane. My field lies at the intersection between the study of geometric structures and the study of discrete subgroups of continuous groups, which generalize the symmetry discussion above. I will investigate a range of examples which are generalizations of the above and work towards solutions to classification problems in those cases. These examples can then be used to formulate general conjectures and theorems. Some of the examples I am interested in have the local geometry of Einstein's static universe, a toy model of spacetime in 3 dimensions. Some more examples have local projective geometry, the type of geometry used in 3D computer graphics. Having a better understanding of these more general notions of geometry and symmetry provides important insight into the fundamental relationship between the two.

从以下意义上讲，流形的局部同质几何结构的领域概括了制图领域：绘制世界的地图集就像试图使用平坦的几何形状来描述地球的形状，而局部均质的几何结构就像试图使用一些同质的几何学来描述更一般的世界（或流形）的形状。主题中的第一个定理之一指出，实际上不可能在没有失真的情况下构建一个完整的世界地图集，即，在二维领域上没有欧几里得（平坦）结构。 Bieberbach定理已经在研究平坦结构的研究中示例了几何结构与对称组之间的紧密关系。这些定理表明，在有限的覆盖层之前，唯一的有限区域平坦结构与Pac-Man的世界相似，也就是说，它们是通过识别平行图的平行边缘而获得的。副图可以通过翻译自身的多个副本，创建具有翻译对称性的瓷砖，这绝非易事。对物体或概念之间对称性的研究是纯数学和其他科学中一系列主题的基础。我们说，如果对象对称性可能是很小的运动，则具有连续的对称性，例如，球体的旋转很小，是对称性的，因此它具有连续的对称性。我们说如果不是这种情况，它具有离散的对称性，例如，立方体具有离散的对称性，因为它只有90度旋转对称性。立方体的对称性是球体对称性的子集。同样，该平面具有连续的对称性，但是平行四边形瓷砖在平面的对称性内具有离散的对称性。我的领域在于几何结构的研究与连续群体的离散亚组的研究之间的相交，这些研究概括了上面的对称讨论。我将调查一系列示例，这些示例是上述概括，并致力于在这些情况下解决分类问题的解决方案。然后可以使用这些示例来制定通用猜想和定理。我感兴趣的一些示例具有爱因斯坦静态宇宙的局部几何形状，这是一个在三个维度上的时空玩具模型。更多示例具有本地投影几何形状，即3D计算机图形中使用的几何类型。对这些几何和对称性的这些更一般的注释有了更好的了解，可以对两者之间的基本关系进行重要的了解。